Chain Rule
The chain rule for differentiating composite functions.
Includes step-by-step examples with nested functions.
The Chain Rule
dy/dx = f'(g(x)) · g'(x)
The chain rule is used to differentiate composite functions — a function inside another function. Think of it as peeling layers: differentiate the outer function first, then multiply by the derivative of the inner function.
In Leibniz Notation
dy/dx = (dy/du) · (du/dx)
Variables
| Symbol | Meaning |
|---|---|
| f(g(x)) | A composite function (outer function f applied to inner function g) |
| f'(g(x)) | Derivative of the outer function, evaluated at g(x) |
| g'(x) | Derivative of the inner function |
Example 1 — Simple Chain Rule
Find d/dx [(3x + 2)⁵]
Outer function: u⁵, Inner function: u = 3x + 2
d/dx [u⁵] = 5u⁴ · du/dx
= 5(3x + 2)⁴ · 3
= 15(3x + 2)⁴
Example 2 — Trigonometric Chain Rule
Find d/dx [sin(x²)]
Outer function: sin(u), Inner function: u = x²
= cos(x²) · d/dx[x²]
= 2x · cos(x²)
Example 3 — Exponential Chain Rule
Find d/dx [e^(4x³)]
Outer function: e^u, Inner function: u = 4x³
= e^(4x³) · d/dx[4x³]
= 12x² · e^(4x³)
Example 4 — Double Chain Rule
Find d/dx [sin²(3x)]
This is [sin(3x)]². Three layers: square → sin → 3x
= 2·sin(3x) · d/dx[sin(3x)]
= 2·sin(3x) · cos(3x) · 3
= 6·sin(3x)·cos(3x) = 3·sin(6x)
When to Use It
Use the chain rule whenever you see:
- A function raised to a power: (something)ⁿ
- A trig function of an expression: sin(something), cos(something)
- An exponential with a non-trivial exponent: e^(something)
- A logarithm of an expression: ln(something)
- Any nested or composite function